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The image presents a table with radical expressions, laws of radicals, and their simplified forms. Let's work through the questions one by one to fill in the missing parts.

### 1. \( \sqrt[4]{\frac{16}{81}} \)
- **Law of Radicals:** Third Law of Radicals (Simplifying fractions under radicals: \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \))
- **Simplified Form:** \( \frac{2}{3} \) (as provided)

### 2. \( \sqrt[3]{(-8)^2} \)
- **Law of Radicals:** Second Law of Radicals (Power under a radical: \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \))
- **Simplified Form:** \( 4 \). This is because \( (-8)^2 = 64 \) and \( \sqrt[3]{64} = 4 \).

### 3. \( \sqrt[5]{1024} \)
- **Law of Radicals:** Second Law of Radicals
- **Simplified Form:** \( 4 \). This is because \( 1024 = 4^5 \), and \( \sqrt[5]{4^5} = 4 \).

### 4. \( \sqrt{50} \)
- **Law of Radicals:** First Law of Radicals (Simplifying square roots: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \))
- **Simplified Form:** \( 5\sqrt{2} \). This is because \( 50 = 25 \times 2 \), and \( \sqrt{25} = 5 \).

### 5. \( \sqrt[3]{x^6} \)
- **Law of Radicals:** Second Law of Radicals
- **Simplified Form:** \( x^2 \). This is because \( \sqrt[3]{x^6} = x^{\frac{6}{3}} = x^2 \).

### 6. \( \sqrt[5]{\frac{5}{40}} \)
- **Law of Radicals:** Third Law of Radicals
- **Simplified Form:** \( \frac{1}{2} \). This is because \( \frac{5}{40} = \frac{1}{8} \), and \( \sqrt[5]{\frac{1}{8}} = \frac{1}{2} \).

Would you like any detailed explanations on any of the steps?

### Follow-up Questions:
1. How do you apply the Second Law of Radicals when simplifying powers under a radical?
2. What is the general rule for simplifying cube roots?
3. How does the Third Law of Radicals help simplify fractions inside radicals?
4. Can you describe how to break down larger numbers into prime factors to simplify square roots?
5. What is the difference between simplifying square roots and cube roots?

**Tip:** Always check if a number under a radical can be factored into perfect squares (or cubes for cube roots) to simplify the expression more easily!

Identify the laws of radicals that exist in the following radical expressions and give the simplified form.

Learn how to simplify various radical expressions using the laws of radicals. This problem involves expressions like cube roots and fifth roots and covers algebraic concepts in simplifying radicals, suitable for Grades 8-10.

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